Subsurface resistivities from anisotropic inversions of electromagnetic data provide quantitative input to hydrocarbon exploration decisions, and it is therefore desirable to have quantitative estimates of the uncertainties associated with those resistivities. Due to the geologic complexity of the subsurface, these resistivities are most accurately determined as three-dimensional models that also capture the effects of anisotropy. In addition to the severe computational burden of working with three-dimensional models and datasets, the quantitative estimation of resistivity uncertainty is hampered by the non-linearity of the inversion process, noisy data, limited bandwidth, and the need to regularize what is otherwise an underdetermined inversion problem. It is desirable to have a method of estimating resistivity uncertainties that can be applied to three-dimensional problems at reasonable computational cost.
There have been two conventional approaches to this problem. The first is to carry out a limited number of sensitivity tests aimed at exposing those features of the subsurface which are most (or least) reliably determined by the data and the inversion process. For example, 3D inversions might be carried out from different starting models or with different degrees of regularization.
The disadvantages of this approach are that it is limited by the number of 3D inversions that can be carried out, relies on somehow identifying the most important sources of uncertainty in a particular problem, and provides only a qualitative view of uncertainty. Occasionally, in an attempt to get a more quantitative description, linearized estimates of the uncertainty are calculated by computing model resolution and covariance matrices; see Alumbaugh and Newman, “Image appraisal for 2-D and 3-D electromagnetic inversion,” (Geophysics 65, 1455-1467 (2000). Although the linearized uncertainty may prove useful for weakly nonlinear problems, one could expect a severe underestimation of the uncertainty model for most of the electromagnetic problems (which are usually highly nonlinear).
The second approach is to approximate the problem, usually by assuming the earth to be one-dimensional, so that the underlying computations become much faster and simpler. This approximation allows for either very large numbers of inversions (deterministic approach) or forward modeling (stochastic approach) to be carried out in an attempt to ensure that important sources of uncertainty are not missed. See “Bayesian inference, Gibbs' sampler and uncertainty estimation in geophysical inversion,” Sen and Stoffa, Geophysical Prospecting 44, 313-350 (1996). While this approach does result in quantitative estimates of uncertainty, the assumption that the earth is laterally invariant strongly limits the value of those estimates.
In an attempt to overcome the deficiencies associated with the two conventional approaches discussed above, Tompkins et al. (“Marine electromagnetic inverse solution appraisal and uncertainty using model-derived basis functions and sparse geometric sampling,” Geophysical Prospecting 59, 947-965 (2011)) suggested a coarser parameterization in a reduced model space defined by means of optimization. This novel approach combines the power of the full 3D anisotropic optimization with a stochastic posterior analysis in a reduced—and therefore tractable—model space. The concept of the reduced model space along with the hybrid nature of the method (both full 3D and stochastic) makes this approach distinctively different from the other two discussed above.